Finally, as remarks we list several possible interesting questions arising from the current paper.
The first is to reinvestigate the coupled KN equation (REF ) and the DNLS equation (REF )
using the double Wronskian structure given in (REF ). Since the couple system ()
is the potential form of the KN\((-1)\)  system (REF ),
it is possible to get solutions for the coupled KN equation (REF ) from \(q=(g/f)_x, ~r=(h/s)_x\)  after
redefining \(\phi , \psi \)  with the dispersion relation of (REF ).
Similar treatment was done in [1]}.
The second is to investigate the FL equations with nonzero backgrounds from bilinear approach.
Note that the assumption (REF ) corresponds to \(q=r=0\)  in the Lax pair.
In addition, it would be interesting to reinvestigate possible analytic domains of wave functions
for the Cauchy problem where \(|u(x,t=0)|^2\)  is algebraic decayed as \(|x|\rightarrow \infty \) .
This was not touched in [2]}, [3]}, [4]}.
Finally, solving the nonlocal FL equation from an analytic approach is also an interesting
problem. Note that the analysis in Sec.REF  implies
soliton solutions arise from the eigenvalue distribution \(\mathbf {H}_N=\mathbf {K}_N^*\) 
or \(\mathbf {H}_N=-\mathbf {K}_N^*\) .
However, interactions of 2SS exhibit more varieties in nonlocal case.
